Myosin and kinesin are very small and powerful motors: molecular machines that are responsible for muscle contraction, cell division, movement of proteins and vesicles, and a variety of other processes in cells. Recent single-molecule experiments on kinesin and myosin have demonstrated that they act in a discrete, stepwise fashion with very high efficiency. The principles that govern their activity and efficiency have been addressed in the conceptual framework of an energy-consuming thermal ratchet. Theoretical understanding of these molecular motors has been within the overdamping approximation of thermal ratchets in which: (1) the inertia (mass) of a molecular motor is completely neglected; (2) the reaction pathways are presumed. The existing theories based on such an approximation, while predicting directed (rectified) movement observed in experiments, fail to agree with experiments quantitatively. In particular, they fail to produce observed high efficiency, a key characteristic of molecular motors. The long-term goal of the proposed research is to fully understand myosin, kinesin, and other molecular motors. The objective of this pilot sub-project application is to study the molecular motors by solving the full thermal ratchet Langevin equation, including inertia and without presuming reaction pathways. The following specific aims will be pursued: 1. Find exact numerical solution to a general thermal ratchet Langevin equation. 2. Find mechanochemical characteristics of kinesin and myosin molecular motors. Numerical solution to the full thermal ratchet Langevin equation is expected to yield better comparison with and better understanding of single-molecule experiments on myosin and kinesin.